Rational points
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Every triple, on one circle
Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.
A fraction on the circle forces a whole point
If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.
The areas a right triangle can have
A right triangle with sides 3, 4 and 5 has area 6; one with sides 3/2, 20/3 and 41/6 has area 5. No right triangle with rational sides has area 1, 2 or 3. Which whole numbers are areas is a thousand-year-old question, and its best answer — two counts of solutions to quadratic equations — rests on one of the great unproved conjectures.
Named alongside it
The objects these essays reach for when they reach for this one.
Pythagorean triplesDescentQuadratic formBijectionCongruent numberConicConjectureElliptic curveExhaustive searchHypotenuseInfinite descentLattice